In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel . Recall that an idempotent morphism is an endomorphism of an object with the property that . Elementary considerations show that every idempotent then has a cokernel. The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories.
Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.
A more complicated example is the category of Chow motives. The construction of Chow motives uses the pseudo-abelian completion described below.
The Karoubi envelope construction associates to an arbitrary category a category together with a functor
such that the image of every idempotent in splits in . When applied to a preadditive category , the Karoubi envelope construction yields a pseudo-abelian category called the pseudo-abelian completion of . Moreover, the functor
is in fact an additive morphism.
To be precise, given a preadditive category we construct a pseudo-abelian category in the following way. The objects of are pairs where is an object of and is an idempotent of . The morphisms
in are those morphisms
such that in . The functor
is given by taking to .
- Artin, 1972, p. 413.
- Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A